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Excitons dynamic in a three-stranded \(\alpha \)-helix protein chains with diagonal and off-diagonal couplings: effects of strong long-range interactions

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Abstract

Exciton dynamics through a three-stranded \(\alpha \)-helix protein chain in the presence of inter-spines and diagonal and off-diagonal couplings as well as long-range interactions are investigated. Its is shown that the behavior of the system is governed by three-coupled modified continuous nonlinear Schrödinger equations. Performing a modulational instability analysis, it appears that both the off-diagonal coupling and the long-range interaction reduce the instability regions and the amplitude of the growth rate. More analytical insights of the system dynamics are obtained by constructing its solutions with the F-expansion method. Many families of solutions are unveiled among them are bright and dark solitary waves, Jacobian elliptic function solutions and hyperbolic function solutions to name just a few. Intensive numerical simulations carried corroborate the analytical solutions found with a good accuracy. In addition, our numerical findings prove that the long-range interactions increase the energy of the waves propagating through the protein chain. From a biological point of view, the model used in the current work better describes the energy transport in protein chains.

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Author information

Authors and Affiliations

  1. Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon

    R. Y. Ondoua

  2. Department of Basic Scientific Teachings, National Advanced School of Engineering of Maroua, University of Maroua, P.O. Box 46, Maroua, Cameroon

    J. C. Mimshe Fewu

  3. Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon

    R. Y. Ondoua, J. C. Mimshe Fewu, D. Belobo Belobo, C. B. Tabi & H. P. Ekobena Fouda

  4. Department of Mathematics and Physical Sciences, National Advanced School of Engineering of Yaoundé, University 12 of Yaounde I, P.O. Box 8390, Yaounde, Cameroon

    D. Belobo Belobo

  5. African Centre for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon

    D. Belobo Belobo

  6. Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana

    C. B. Tabi

Authors
  1. R. Y. Ondoua
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  2. J. C. Mimshe Fewu
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  3. D. Belobo Belobo
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  4. C. B. Tabi
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  5. H. P. Ekobena Fouda
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Corresponding author

Correspondence to R. Y. Ondoua.

Appendix: coefficients of Eq. (21)

Appendix: coefficients of Eq. (21)

$$\begin{aligned} \theta _{5}&=m_{{55}}-m_{{22}}+m_{{44}}+m_{{66}}-m_{{33}}-m_{{11}} \\ \theta _{4}&= m_{{11}}m_{{44}}+m_{{11}}m_{{66}}-m_{{22}}m_{{33}}+m_{{22}}m_{{55}}-m_ {{11}}m_{{33}} \\&\quad +m_{{11}}m_{{55}}+m_{{33}}m_{{66}}-{m_{{14}}}^{2}+m_{{22 }}m_{{44}}\\&\quad -m_{{22}}m_{{11}}-{m_{{25}}}^{2}+m_{{33}}m_{{44}}-m_{{55}}m_{{66}}-{m_{{36}}}^{2} \\&\quad +m_{{22}}m_{{66}} -m_{{44}}m_{{55}}-m_{{44}}m_{{66}}+m_{{33}}m_{{55}}, \\ \theta _{3}&=m_{{22}}m_{{33}}m_{{66}}+m_{{22}}m_{{33}}m_{{55}}+m_{{22}}m_{{11}}m_{{ 44}}-m_{{22}}m_{{44}}m_{{66}}\\&\quad +m_{{22}}m_{{11}}m_{{55}}-m_{{33}}m_{{44} }m_{{55}}\\&\quad +m_{{44}}m_{{55}}m_{{66}}-{m_{{25}}}^{2}m_{{11}} -{m_{{14}}}^{2}m_{{22}}+{m_{{25}}}^{2}m_{{44}}+{m_{{25}}}^{2}m_{{66}} \\&\quad +{m_{{14}}}^{2 }m_{{55}}+{m_{{14}}}^{2}m_{{66}}-{m_{{14}}}^{2}m_{{33}}\\&\quad +{m_{{36}}}^{2}m_{{55}}+{m_{{36}}}^{2}m_{{44}}-m_{{22}}{m_{{36}}}^{2}-{m_{{25}}}^{2}m _{{33}}\\&\quad -m_{{11}}{m_{{36}}}^{2}-m_{{11}}m_{{55}}m_{{66}}-m_{{11}}m_{{44 }}m_{{66}}\\&\quad -m_{{33}}m_{{44}}m_{{66}}+m_{{11}}m_{{33}}m_{{66}}-m_{{11}}m _{{44}}m_{{55}}+m_{{11}}m_{{33}}m_{{44}}\\&\quad +m_{{11}}m_{{33}}m_{{55}}-m_{{ 22}}m_{{55}}m_{{66}}\\&\quad -m_{{22}}m_{{11}}m_{{33}}+m_{{22}}m_{{11}}m_{{66}} -m_{{33}}m_{{55}}m_{{66}}-m_{{22}}m_{{44}}m_{{55}}+m_{{22}}m_{{33}}m_{ {44}}, \\ \theta _{2}&=-m_{{11}}m_{{33}}m_{{44}}m_{{55}}+m_{{22}}m_{{11}}m_{{33}}m_{{44}}+m_{ {22}}m_{{11}}m_{{33}}m_{{55}} \\&\quad -m_{{22}}m_{{11}}m_{{44}}m_{{66}}-m_{{22} }m_{{11}}m_{{55}}m_{{66}}\\&\quad +m_{{33}}m_{{44}}m_{{55}}m_{{66}}+m_{{11}}m_{ {44}}m_{{55}}m_{{66}}-m_{{22}}m_{{11}}m_{{44}}m_{{55}} \\&\quad -m_{{22}}m_{{33} }m_{{55}}m_{{66}}-m_{{22}}m_{{33}}m_{{44}}m_{{66}}\\&\quad +m_{{22}}m_{{44}}m_{ {55}}m_{{66}}+{m_{{25}}}^{2}m_{{11}}m_{{66}}-{m_{{25}}}^{2}m_{{33}}m_{ {11}}+{m_{{25}}}^{2}m_{{33}}m_{{44}} \\&\quad -{m_{{14}}}^{2}m_{{22}}m_{{33}}+{m _{{12}}}^{2}m_{{22}}m_{{44}}\\&\quad -2\,m_{{25}}m_{{33}}{m_{{23}}}^{2}+{m_{{25 }}}^{2}m_{{44}}m_{{11}}-{m_{{25}}}^{2}m_{{44}}m_{{66}}+{m_{{13}}}^{2}m _{{33}}m_{{44}} \\&\quad +{m_{{25}}}^{2}m_{{33}}m_{{66}}+{m_{{12}}}^{2}m_{{44}}m _{{55}}\\&\quad -2\,{m_{{12}}}^{2}m_{{44}}m_{{25}}-2\,{m_{{12}}}^{2}m_{{14}}m_{ {55}}+m_{{11}}{m_{{36}}}^{2}m_{{55}}-2\,{m_{{12}}}^{2}m_{{22}}m_{{14}} \\&\quad -{m_{{14}}}^{2}m_{{55}}m_{{66}}+{m_{{14}}}^{2}m_{{22}}m_{{55}}\\&\quad +{m_{{14 }}}^{2}m_{{22}}m_{{66}}+{m_{{14}}}^{2}m_{{33}}m_{{55}}+m_{{22}}m_{{11} }m_{{33}}m_{{66}} \\&\quad -m_{{22}}m_{{33}}m_{{44}}m_{{55}}-2\,{m_{{13}}}^{2}m_ {{33}}m_{{14}}\\&\quad +{m_{{13}}}^{2}m_{{33}}m_ {{11}}+4\,{m_{{12}}}^{2}m_{{14}}m_{{25}}+m_{{22}}m_{{33}}{m_{{23}}}^{2 }-{m_{{36}}}^{2}m_{{44}}m_{{55}} \\&\quad +m_{{22}}{m_{{23}}}^{2}m_{{66}}+{m_{{ 12}}}^{2}m_{{22}}m_{{11}}\\&\quad +m_{{11}}{m_{{36}}}^{2}m_{{44}}-2\,m_{{22}}{m _{{23}}}^{2}m_{{36}}+{m_{{12}}}^{2}m_{{11}}m_{{55}}-2\,{m_{{12}}}^{2}m _{{11}}m_{{25}} \\&\quad +m_{{22}}{m_{{36}}}^{2}m_{{55}}-2\,{m_{{13}}}^{2}m_{{36 }}m_{{44}}\\&\quad +{m_{{13}}}^{2}m_{{11}}m_{{66}}+4\,{m_{{13}}}^{2}m_{{36}}m_{ {14}}-2\,{m_{{13}}}^{2}m_{{36}}m_{{11}} \\&\quad -2\,{m_{{13}}}^{2}m_{{14}}m_{{ 66}}-m_{{11}}m_{{33}}m_{{44}}m_{{66}}\\&\quad +4\,{m_{{23}}}^{2}m_{{36}}m_{{25} }+{m_{{13}}}^{2}m_{{44}}m_{{66}}-2\,{m_{{23}}}^{2}m_{{36}}m_{{55}}-2\, {m_{{23}}}^{2}m_{{25}}m_{{66}} \\&\quad +{m_{{23}}}^{2}m_{{55}}m_{{66}}+m_{{22}} {m_{{36}}}^{2}m_{{44}}\\&\quad -m_{{22}}m_{{11}}{m_{{36}}}^{2}-m_{{11}}m_{{33}} m_{{55}}m_{{66}}-{m_{{14}}}^{2}{m_{{36}}}^{2} \\&\quad -{m_{{25}}}^{2}{m_{{36}}} ^{2}+{m_{{14}}}^{2}m_{{33}}m_{{66}}\\&\quad -{m_{{25}}}^{2}{m_{{14}}}^{2}+{m_{{23}}}^{2}m_{{33}}m_{{55}}, \\ \theta _{1}&=4\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_{{25}}-{m_{{12}}}^{2}m_{{44}}m_{{55 }}m_{{66}} \\&\quad +2\,{m_{{12}}}^{2}m_{{44}}m_{{25}}m_{{66}}+2\,{m_{{12}}}^{2} m_{{14}}m_{{55}}m_{{66}}\\&\quad -4\,{m_{{12}}}^{2}m_{{14}}m_{{25}}m_{{66}}-2\, m_{{25}}m_{{33}}{m_{{23}}}^{2}m_{{11}}+2\,m_{{25}}m_{{33}}m_{{44}}{m_{ {23}}}^{2} \\&\quad -{m_{{25}}}^{2}m_{{33}}{m_{{14}}}^{2}+{m_{{25}}}^{2}{m_{{36} }}^{2}m_{{44}}\\&\quad -{m_{{14}}}^{2}m_{{22}}{m_{{36}}}^{2}-2\,{m_{{12}}}^{2}m _{{33}}m_{{44}}m_{{25}}+{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{44}} \\&\quad +{m_{{14 }}}^{2}m_{{22}}m_{{33}}m_{{55}}-{m_{{14}}}^{2}m_{{22}}m_{{55}}m_{{66}} \\&\quad -{m_{{25}}}^{2}m_{{33}}m_{{44}}m_{{66}}+{m_{{25}}}^{2}m_{{33}}m_{{11}} m_{{66}} \\&\quad +{m_{{25}}}^{2}m_{{33}}m_{{44}}m_{{11}}-{m_{{25}}}^{2}m_{{44}} m_{{11}}m_{{66}}\\&\quad -m_{{22}}m_{{ 11}}m_{{33}}m_{{55}}m_{{66}}-m_{{22}}m_{{11}}m_{{33}}m_{{44}}m_{{66}}- m_{{22}}m_{{11}}m_{{33}}m_{{44}}m_{{55}} \\&\quad +m_{{22}}m_{{11}}m_{{44}}m_{{ 55}}m_{{66}}\\&\quad +m_{{22}}m_{{33}}m_{{44}}m_{{55}}m_{{66}}+{m_{{12}}}^{2}m_ {{33}}m_{{44}}m_{{55}} \\&\quad +m_{{22}}m_{{33}}{m_{{23}}}^{2}m_{{11}}-m_{{22}} m_{{33}}m_{{44}}{m_{{23}}}^{2}\\&\quad -{m_{{23}}}^{2}m_{{33}}m_{{44}}m_{{55}}+2\,{m_{{13}}}^{2}m_{{33}}m_{{14 }}m_{{55}} \\&\quad -m_{{22}}{m_{{23}}}^{2}m_{{44}}m_{{66}}-2\,m_{{22}}{m_{{23}} }^{2}m_{{11}}m_{{36}}\\&\quad +2\,m_{{22 }}{m_{{23}}}^{2}m_{{44}}m_{{36}}-2\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_{{ 55}}+{m_{{12}}}^{2}m_{{33}}m_{{11}}m_{{55}} \\&\quad +2\,{m_{{23}}}^{2}m_{{36}}m _{{44}}m_{{55}}\\&\quad -4\,{m_{{23}}}^{2}m_{{36}}m_{{44}}m_{{25}}-{m_{{13}}}^{ 2}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{13}}}^{2}m_{{14}}m_{{55}}m_{{66}} \\&\quad -{ m_{{13}}}^{2}m_{{11}}m_{{55}}m_{{66}} \\&\quad -4\,{m_{{13}}}^{2}m_{{36}}m_{{14} }m_{{55}}+2\,{m_{{13}}}^{2}m_{{36}}m_{{11}}m_{{55}} \\&\quad +2\,{m_{{13}}}^{2}m _{{36}}m_{{44}}m_{{55}}+{m_{{13}}}^{2}m_{{22}}m_{{44}}m_{{66}}\\&\quad -{m_{{13 }}}^{2}m_{{33}}m_{{11}}m_{{55}} -{m_{{13}}}^{2}m_{{33}}m_{{44}}m_{{55}} -2\,{m_{{13}}}^{2}m_{{22}}m_{{33}}m_{{14}} \\&\quad +{m_{{13}}}^{2}m_{{22}}m_{{ 33}}m_{{11}}\\&\quad +2\,{m_{{12}}}^{2}m _{{22}}m_{{14}}m_{{66}}-{m_{{12}}}^{2}m_{{22}}m_{{11}}m_{{66}}-{m_{{12 }}}^{2}m_{{22}}m_{{44}}m_{{66}} \\&\quad -2\,{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{ 14}}\\&\quad -2\,{m_{{12}}}^{2}m_{{33}}m_{{11}}m_{{25}}-{m_{{12}}}^{2}m_{{11}}m _{{55}}m_{{66}} \\&\quad +2\,{m_{{12}}}^{2}m_{{11}}m_{{25}}m_{{66}}-m_{{11}}{m_{ {36}}}^{2}m_{{44}}m_{{55}}\\&\quad +m_{{ 22}}m_{{11}}{m_{{36}}}^{2}m_{{44}}-m_{{22}}{m_{{36}}}^{2}m_{{44}}m_{{ 55}}+{m_{{23}}}^{2}m_{{11}}m_{{55}}m_{{66}} \\&\quad -2\,{m_{{23}}}^{2}m_{{11}}m _{{25}}m_{{66}}\\&\quad +2\,{m_{{23}}}^{ 2}m_{{44}}m_{{25}}m_{{66}}-2\,{m_{{23}}}^{2}m_{{36}}m_{{55}}m_{{11}}+4 \,{m_{{23}}}^{2}m_{{36}}m_{{25}}m_{{11}} \\&\quad +{m_{{14}}}^{2}m_{{22}}m_{{33} }m_{{66}}\\&\quad +{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{11}}-2\,{m_{{13}}}^{2}m_{{ 22}}m_{{14}}m_{{66}}+{m_{{13}}}^{2}m_{{22}}m_{{11}}m_{{66}} \\&\quad +4\,{m_{{13 }}}^{2}m_{{22}}m_{{36}}m_{{14}}\\&\quad -2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{ 11}}-2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{44}}-{m_{{14}}}^{2}m_{{33}}m _{{55}}m_{{66}} \\&\quad +{m_{{14}}}^{2}{m_{{36}}}^{2}m_{{55}}+{m_{{25}}}^{2}{m_ {{14}}}^{2}m_{{66}}\\&\quad -{m_{{25}}}^{2}{m_{{36}}}^{2}m_{{11}}-{m_{{23}}}^{2}m_{{44}}m_{{55}}m_{{66}} +m_{{22}}m_{{11}}{m_{{36}}}^{2}m_{{55}} \\&\quad +{m_{{13}}}^{2}m_{{22}}m_{{33}}m_{{44}} +m_{{22}}{m_{{23}}}^{2}m_{{11}}m_{{66}}\\&\quad +{m_{{23}}}^{2}m_{{33}}m_{{55}}m_{{11}}+m_{{11}}m_{{33}}m_{{44}}m_{{55}}m_{{66}}, \\ \theta _{0}&=-m_{{22}}{m_{{23}}}^{2}m_{{11}}m_{{44}}m_{{66}}-4\,m_{{12}}m_{{33}}m_{ {14}}m_{{55}}m_{{23}}m_{{13}} +8\,m_{{12}}m_{{33}}m_{{14}}m_{{25}}m_{{ 23}}m_{{13}}\\&\quad -2\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{44}}m_{{25}}+2\,m_{{23 }}m_{{13}}m_{{44}}m_{{55}}m_{{12}}m_{{66}} \\&\quad +{m_{{36}}}^{2}{m_{{12}}}^{2 }m_{{44}}m_{{55}}-2\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{14}}m_{{55}}\\&\quad -4\,m _{{23}}m_{{13}}m_{{44}}m_{{25}}m_{{12}}m_{{66}}-m_{{22}}m_{{33}}m_{{44 }}{m_{{23}}}^{2}m_{{11}}-{m_{{23}}}^{2}m_{{33}}m_{{44}}m_{{55}}m_{{11} } \\&\quad +4\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{14}}m_{{25}}\\&\quad +{m_{{36}}}^{2}{m_{{ 12}}}^{2}m_{{11}}m_{{55}}-2\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{11}}m_{{ 25}}-4\,m_{{23}}m_{{13}}m_{{14}}m_{{55}}m_{{12}}m_{{66}} \\&\quad -2\,{m_{{25}}} ^{2}{m_{{13}}}^{2}m_{{14}}m_{{66}}\\&\quad -4\,m_{{36}}m_{{12}}m_{{23}}m_{{13}} m_{{44}}m_{{55}}+8\,m_{{36}}m_{{12}}m_{{23}}m_{{13}}m_{{44}}m_{{25}} \\&\quad +8 \,m_{{36}}m_{{12}}m_{{23}}m_{{13}}m_{{14}}m_{{55}}\\&\quad -16\,m_{{36}}m_{{12} }m_{{23}}m_{{13}}m_{{14}}m_{{25}}+2\,m_{{12}}m_{{22}}m_{{33}}m_{{44}}m _{{23}}m_{{13}}-2\,m_{{22}}{m_{{23}}}^{2}{m_{{14}}}^{2}m_{{36}}\\&\quad +{m_{{23}}}^{2}m_{{33}}{m_{{14}} }^{2}m_{{55}}+8\,m_{{23}}m_{{13}}m_{{14}}m_{{25}}m_{{12}}m_{{66}} \\&\quad -4\,{ m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{14}}m_{{55}}\\&\quad +2\,m_{{12}}m_{{22}}m_{{ 23}}m_{{13}}m_{{11}}m_{{66}}-4\,m_{{12}}m_{{22}}m_{{13}}m_{{11}}m_{{23 }}m_{{36}}+2\,{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{14}}m_{{66}}\\&\quad +{m_{{25}} }^{2}m_{{33}}{m_{{14}}}^{2}m_{{66}}-2\,{m_{{25}}}^{2}m_{{33}}m_{{14}}{ m_{{13}}}^{2}+{m_{{25}}}^{2}m_{{33}}m_{{11}}{m_{{13}}}^{2} \\&\quad +{m_{{25}}}^ {2}m_{{33}}m_{{44}}{m_{{13}}}^{2}\\&\quad -2\,{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{ 11}}m_{{36}}+{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{11}}m_{{66}}+{m_{{25}}}^{ 2}{m_{{36}}}^{2}m_{{44}}m_{{11}} \\&\quad +m_{{22}}m_{{33}}{m_{{14}}}^{2}{m_{{23 }}}^{2}\\&\quad -{m_{{12}}}^{2}m_{{33}}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{12}}}^{ 2}m_{{33}}m_{{44}}m_{{25}}m_{{66}} \\&\quad +2\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_ {{55}}m_{{66}}\\&\quad -{m_{{ 12}}}^{2}m_{{33}}m_{{11}}m_{{55}}m_{{66}}+2\,{m_{{12}}}^{2}m_{{33}}m_{ {11}}m_{{25}}m_{{66}}-{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{11}}m_{{66}} \\&\quad -{ m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{44}}m_{{66}}\\&\quad +2\,m_{{12}}m_{{33}}m_{{ 44}}m_{{55}}m_{{23}}m_{{13}}+2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{11}} m_{{55}} \\&\quad +2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{44}}m_{{55}}\\&\quad -{m_{{14}}}^ {2}m_{{22}}m_{{33}}m_{{55}}m_{{66}}-{m_{{25}}}^{2}m_{{33}}m_{{44}}m_{{ 11}}m_{{66}}+2\,m_{{25}}m_{{33}}m_{{44}}{m_{{23}}}^{2}m_{{11}} \\&\quad -2\,{m_{ {12}}}^{2}m_{{22}}{m_{{36}}}^{2}m_{{14}}\\&\quad +{m_{{12}}}^{2}m_{{22}}{m_{{36 }}}^{2}m_{{11}}+{m_{{12}}}^{2}m_{{22}}{m_{{36}}}^{2}m_{{44}}+2\,{m_{{ 13}}}^{2}m_{{22}}m_{{33}}m_{{14}}m_{{55}} \\&\quad -{m_{{13}}}^{2}m_{{22}}m_{{33 }}m_{{11}}m_{{55}}\\&\quad -{m_{{13}}}^{2}m_{{22}}m_{{33}}m_{{44}}m_{{55}}-{m_{ {25}}}^{2}{m_{{36}}}^{2}{m_{{14}}}^{2} \\&\quad -4\,m_{{12}}m_{{22}}m_{{13}}m_{{ 44}}m_{{23}}m_{{36}}\\&\quad +8\,m_{{12}}m_{{22}}m_{{13}}m_{{14}}m_{{23}}m_{{36}}-4\,m_{{12}}m_{{ 22}}m_{{13}}m_{{14}}m_{{23}}m_{{66}} \\&\quad -4\,m_{{12}}m_{{22}}m_{{33}}m_{{14 }}m_{{23}}m_{{13}}\\&\quad +2\,m_{{12}}m_{{22}}m_{{33}}m_{{11}}m_{{23}}m_{{13}} -4\,m_{{12}}m_{{33}}m_{{44}}m_{{25}}m_{{23}}m_{{13}} \\&\quad +{m_{{23}}}^{2}{m_ {{14}}}^{2}m_{{55}}m_{{66}}\\&\quad -{m_{{23}}}^{2}m_{{11}}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{23}}}^{2}m _{{11}}m_{{44}}m_{{25}}m_{{66}} \\&\quad +2\,{m_{{23}}}^{2}m_{{36}}m_{{44}}m_{{ 55}}m_{{11}}\\&\quad -{m_{{13 }}}^{2}m_{{22}}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{13}}}^{2}m_{{22}}m_{{ 14}}m_{{55}}m_{{66}} \\&\quad -{m_{{13}}}^{2}m_{{22}}m_{{11}}m_{{55}}m_{{66}} \\&\quad +m_{{22}}m_{{11}}m_{{ 33}}m_{{44}}m_{{55}}m_{{66}}-2\,{m_{{23}}}^{2}{m_{{14}}}^{2}m_{{25}}m_ {{66}}-2\,{m_{{23}}}^{2}m_{{36}}{m_{{14}}}^{2}m_{{55}} \\&\quad +4\,{m_{{23}}}^{ 2}m_{{36}}{m_{{14}}}^{2}m_{{25}}\\&\quad + {m_{{14}}}^{2}m_{{22}}{m_{{36}}}^{2}m _{{55}}-2\,m_{{25}}m_{{33}}{m_{{14}}}^{2}{m_{{23}}}^{2}-2\,{m_{{25}}}^ {2}{m_{{13}}}^{2}m_{{44}}m_{{36}} \\&\quad +{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{44}} m_{{66}}\\&\quad +4\,{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{14}}m_{{36}}+2\,m_{{12}}m_ {{33}}m_{{11}}m_{{55}}m_{{23}}m_{{13}} \\&\quad -4\,m_{{12}}m_{{33}}m_{{11}}m_{{ 25}}m_{{23}}m_{{13}}\\&\quad +2\,m_{{23}}m_{{13}}m_{{11}}m_{{55}}m_{{12}}m_{{66 }}-4\,m_{{23}}m_{{13}}m_{{11}}m_{{25}}m_{{12}}m_{{66}} \\&\quad -4\,m_{{36}}m_{{ 12}}m_{{23}}m_{{13}}m_{{11}}m_{{55}}\\&\quad +8\,m_{{36}}m_{{12}}m_{{23}}m_{{13 }}m_{{11}}m_{{25}}-4\,{m_{{23}}}^{2}m_{{36}}m_{{44}}m_{{25}}m_{{11}} \\&\quad -m_{{22}}m_{{11}}{m_{{36}}}^{2}m_{{44}}m_{{55}} \\&\quad +2\,m_{{22}}{m_{{23}}}^{2}m_{{11}}m_{{44}}m_{{36}}+2\,m_{{12}}m_{{22}}m_{{13}}m_{{44}}m_{{23}}m_{{66 }} \\&\quad -4\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_{{25}}m_{{66}}\\&\quad +m_{{22}}{m_{{23}}}^{2}{m_{{14}}}^{2}m_{{66}}. \end{aligned}$$

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Ondoua, R.Y., Mimshe Fewu, J.C., Belobo Belobo, D. et al. Excitons dynamic in a three-stranded \(\alpha \)-helix protein chains with diagonal and off-diagonal couplings: effects of strong long-range interactions. Eur. Phys. J. Plus 136, 274 (2021). https://doi.org/10.1140/epjp/s13360-021-01279-1

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  • DOI: https://doi.org/10.1140/epjp/s13360-021-01279-1

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